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Physical Review A

Initiation of superfluorescence from a sphere

Jan Mostowski, Božena Sobolewska

Published Nov 1, 1983• 28 citations
Publisher

Abstract

Three-dimensional theory of the initiation of superfluorescence from a spherical medium with a radius much larger than the optical wavelength is developed. Time evolution of the average intensity of the superfluorescence pulse from its initial spontaneous stage to the exponential-growth limit is found. It is shown that light propagates along rays, governed by laws of geometrical optics. Amplification of light along each ray is the same as determined from the conventional one-dimensional theory. It is proved that although the average intensity is isotropic, each pulse is strongly directional.

DOI
10.1103/physreva.28.2943

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Three-dimensional theory of the initiation of superfluorescence from a spherical medium with a radius much larger than the optical wavelength is developed. Time evolution of the average intensity of the superfluorescence pulse from its initial spontaneous stage to the exponential-growth limit is found. It is shown that light propagates along rays, governed by laws of geometrical optics. Amplification of light along each ray is the same as determined from the conventional one-dimensional theory. It is proved that although the average intensity is isotropic, each pulse is strongly directional.
We investigate the early stages of superfluorescent emission from a large spherical volume containing atoms all initially in the same excited state. The electric polarization fluctuations that characterize a fully inverted atomic system, and which trigger superfluorescence, are found to cause field amplification only in vector spherical-harmonic multipole modes of limited order. The field and polarization fluctuations obey Gaussian statistics, a result that has been used to calculate the angular correlation of intensities radiated in any two directions. Components of the electric field radiated in two arbitrary directions and circularly polarized in opposite senses are found to be uncorrelated. They thus have no intensity-intensity or higher-order correlations. Field components which are circularly polarized in the same sense, however, are shown to be correlated over an angular range that is proportional to the ratio of the wavelength to the radius of the sphere.
We have formulated the quantum theory of formation of the superfluorescence pulse from a pencil-like amplifier. The coupled system of the linearized Bloch equations for two-level atoms and the three-dimensional wave equation for the electric field has been solved in the geometrical-optics approximation for Fresnel numbers $F \ensuremath{\ge} 1$. Several properties of the superfluorescence pulse are discussed. The spatial coherence function of the pulse has been found as a function of the Fresnel number. This function provides a precise formulation of the concept of "diffraction modes." The delay-time distribution function for various Fresnel numbers has been found. The results are consistent with experimental data and numerical simulations.
A model is presented for the dynamical evolution of superfluorescence from an optically pumped three-level system. The full propagation, transverse, and diffraction effects are taken into account. With the use of a previously developed algorithm, a computational simulation was constructed from this model and results are presented and discussed. In particular, it is shown that the injected coherent pump-pulse initial characteristics, such as on-axis area, temporal and radial width and shape, can have significant deterministic effects on the superfluorescent pulse delay time, peak intensity, temporal width, and shape. Thus, by specifying certain initial properties of the injected pump pulse, the superfluorescent pulse can be shaped and altered. The results predict the conditions under which an injected light pulse of a given frequency can be used to generate, shape, and control a second light pulse of a different frequency via a nonlinear medium, thus demonstrating a new aspect of the phenomenon of light control by light.
Conservation laws for Maxwell-Bloch equations applied to describe superfluorescence are studied. Modifications due to the initial boundary conditions are discussed. For initial polarization in the form of a constant tipping angle, conservation laws are integrated to give exact dependence of all field moments on the propagation length. Three regions of propagation are distinguished: the linear region, where the field energy is negligibly small, the region of rapid exponential growth, and the saturation region. The characteristic distance, where maximum radiation occurs, is explicitly evaluated; it defines the threshold length for superfluoresence. A small chirping, proportional to the square of the tipping angle, is found. The change of the pulse width with propagation length is estimated.
An analysis is made of the conditions for the generation of superfluorescence pulses in an inverted medium of electron—hole pairs in a semiconductor. It is shown that strong optical amplification in laser semiconductor amplifiers characterised by α L ≫ 1 (α is the small-signal gain and L is the amplifier length) leads to suppression of phase relaxation of the medium during the initial stages of evolution of superfluorescence and to formation of a macroscopic dipole from electron—hole pairs. Cooperative emission of radiation in this system results in generation of a powerful ultrashort pulse of the optical gain, which interacts coherently with the semiconductor medium. It is shown that coherent pulsations of the optical field, observed earlier by the author in Q -switched semiconductor lasers, are the result of superfluorescence and of the coherent interaction between the optical field and the medium.
An approach to modeling the dynamics of x-ray amplified spontaneous emission and superfluorescence, the phenomenon of collective x-ray emission initiated by intense pulses of x-ray free-electron lasers, is developed based on stochastic partial differential equations. The equations are derived from first principles, and the relevant approximations, derivation steps, and extensions specific to stimulated x-ray emission are presented. The resulting equations take the form of three-dimensional generalized Maxwell-Bloch equations augmented with noise terms for both field and atomic variables. The derived noise terms possess specific correlation properties that enable the correct reconstruction of spontaneous emission. Consequently, the developed theoretical formalism is universally suitable for describing all stages of stimulated x-ray emission: spontaneous emission, amplified spontaneous emission, and superfluorescence. We present numerical examples that illustrate various properties of the emitted field, including spatiotemporal coherence and spectral-angular and polarization characteristics. We anticipate that the proposed theoretical framework will establish a robust foundation for interpreting measurements in stimulated x-ray emission spectroscopy, modeling x-ray laser oscillators, and describing other experiments leveraging x-ray superfluorescence.